Datasets:

roseking
/

openAI-math-dataset

import OAI.AlgebraicGeometry.StableCoordinate.DeterminantRing
import OAI.AlgebraicGeometry.StableCoordinate.RigidityCoefficients
import OAI.AlgebraicGeometry.StableCoordinate.FactorialClosure
import Mathlib.Algebra.CharP.Algebra

namespace OAI

/-!
# Rigidity helpers on the actual determinant quotient

These results instantiate the general derivation arguments on the exact ring
`k[a,d,b,c,u]/(ac-bd-1)`.  Generator nonvanishing is proved by an algebraic
point, not assumed or inferred from an unproved localization identification.
-/

noncomputable section

namespace StableCoordinate.DeterminantRing

open MvPolynomial

variable (k : Type*) [CommRing k]

/-- The distinguished element of source equation `bun:v`. -/
def v : C k := a k ^ 3 * b k + a k ^ 2 * u k ^ 2 - d k ^ 2

/-- A point at which the factors used by the rigidity argument are nonzero. -/
def factorPoint : C k →ₐ[k] k := lift 1 0 0 1 1 (by simp)

@[simp] theorem factorPoint_a : factorPoint k (a k) = 1 := by simp [factorPoint]
@[simp] theorem factorPoint_d : factorPoint k (d k) = 0 := by simp [factorPoint]
@[simp] theorem factorPoint_u : factorPoint k (u k) = 1 := by simp [factorPoint]

/-- The scalar inclusion is injective, witnessed by `factorPoint`. -/
theorem algebraMap_injective : Function.Injective (algebraMap k (C k)) := by
  intro x y h
  have h' := congrArg (factorPoint k) h
  simpa only [AlgHom.commutes, Algebra.algebraMap_self, RingHom.id_apply] using h'

instance instCharZero [CharZero k] : CharZero (C k) :=
  charZero_of_injective_algebraMap (algebraMap_injective k)

instance instNontrivial [Nontrivial k] : Nontrivial (C k) :=
  Function.Injective.nontrivial (algebraMap_injective k)

/-- The first base variable is nonzero in the actual quotient. -/
theorem a_ne_zero [Nontrivial k] : a k ≠ 0 := by
  intro h
  have := congrArg (factorPoint k) h
  simp at this

/-- The added base variable is nonzero in the actual quotient. -/
theorem u_ne_zero [Nontrivial k] : u k ≠ 0 := by
  intro h
  have := congrArg (factorPoint k) h
  simp at this

/-- The two factors of the highest auxiliary part of `v` are nonzero. -/
theorem au_sub_d_ne_zero [Nontrivial k] : a k * u k - d k ≠ 0 := by
  intro h
  have := congrArg (factorPoint k) h
  simp at this

theorem au_add_d_ne_zero [Nontrivial k] : a k * u k + d k ≠ 0 := by
  intro h
  have := congrArg (factorPoint k) h
  simp at this

/-- Every element of the determinant ring has a polynomial representative. -/
theorem mk_surjective : Function.Surjective (mk k) :=
  Ideal.Quotient.mk_surjective

/-- The five canonical quotient variables generate the full algebra. -/
theorem adjoin_range_generator_via_quotient :
    Algebra.adjoin k (Set.range (generator k)) = ⊤ := by
  have h : (Algebra.adjoin k (Set.range (X : Fin 5 → P k))).map (mk k) = ⊤ := by
    rw [MvPolynomial.adjoin_range_X, Algebra.map_top]
    exact (AlgHom.range_eq_top (mk k)).mpr (mk_surjective k)
  change Algebra.adjoin k (Set.range fun i : Fin 5 => mk k (X i)) = ⊤
  simpa only [AlgHom.map_adjoin, ← Set.range_comp, Function.comp_def] using h

/-- A derivation is determined by its five generator values. -/
theorem derivation_eq_zero (D : Derivation k (C k) (C k))
    (ha : D (a k) = 0) (hd : D (d k) = 0) (hb : D (b k) = 0)
    (hc : D (c k) = 0) (hu : D (u k) = 0) : D = 0 := by
  apply DFunLike.ext
  intro z
  obtain ⟨p, rfl⟩ := mk_surjective k z
  change D (mk k p) = 0
  induction p using MvPolynomial.induction_on with
  | C r =>
      change D (algebraMap k (C k) r) = 0
      exact D.map_algebraMap r
  | add p q hp hq => simp only [map_add, hp, hq, add_zero]
  | mul_X p i hp =>
      rw [map_mul, D.leibniz, hp, smul_zero, add_zero]
      have hi : D (mk k (X i)) = 0 := by
        fin_cases i
        · exact ha
        · exact hd
        · exact hb
        · exact hc
        · exact hu
      rw [hi, smul_zero]

/-- The coefficient of a nonzero derivation fixing `a,d,u` is nonzero. -/
theorem coefficient_ne_zero (D : Derivation k (C k) (C k))
    (hD : D ≠ 0) (ha : D (a k) = 0) (hd : D (d k) = 0) (hu : D (u k) = 0) :
    rigidityCoefficient D (b k) (c k) ≠ 0 := by
  intro hq
  obtain ⟨hb, hc⟩ := derivation_rigidityCoefficient D (a k) (d k) (b k) (c k)
    (determinant_relation k) ha hd
  rw [hq, mul_zero] at hb hc
  exact hD (derivation_eq_zero k D ha hd hb hc hu)

/-- The difference-of-squares argument instantiated on the actual quotient. -/
theorem fixed_of_difference_of_squares [Nontrivial k] [IsDomain (C k)] [CharZero (C k)]
    (D : Derivation k (C k) (C k)) (hD : IsLocallyNilpotent D)
    (hv : D (a k ^ 2 * u k ^ 2 - d k ^ 2) = 0) :
    D (a k) = 0 ∧ D (d k) = 0 ∧ D (u k) = 0 :=
  lnd_fixed_of_difference_of_squares D hD (a_ne_zero k) (u_ne_zero k)
    (au_sub_d_ne_zero k) (au_add_d_ne_zero k) hv

end StableCoordinate.DeterminantRing

end

end OAI

	import OAI.AlgebraicGeometry.StableCoordinate.DeterminantRing
	import OAI.AlgebraicGeometry.StableCoordinate.RigidityCoefficients
	import OAI.AlgebraicGeometry.StableCoordinate.FactorialClosure
	import Mathlib.Algebra.CharP.Algebra

	namespace OAI

	/-!
	# Rigidity helpers on the actual determinant quotient

	These results instantiate the general derivation arguments on the exact ring
	`k[a,d,b,c,u]/(ac-bd-1)`. Generator nonvanishing is proved by an algebraic
	point, not assumed or inferred from an unproved localization identification.
	-/

	noncomputable section

	namespace StableCoordinate.DeterminantRing

	open MvPolynomial

	variable (k : Type*) [CommRing k]

	/-- The distinguished element of source equation `bun:v`. -/
	def v : C k := a k ^ 3 * b k + a k ^ 2 * u k ^ 2 - d k ^ 2

	/-- A point at which the factors used by the rigidity argument are nonzero. -/
	def factorPoint : C k →ₐ[k] k := lift 1 0 0 1 1 (by simp)

	@[simp] theorem factorPoint_a : factorPoint k (a k) = 1 := by simp [factorPoint]
	@[simp] theorem factorPoint_d : factorPoint k (d k) = 0 := by simp [factorPoint]
	@[simp] theorem factorPoint_u : factorPoint k (u k) = 1 := by simp [factorPoint]

	/-- The scalar inclusion is injective, witnessed by `factorPoint`. -/
	theorem algebraMap_injective : Function.Injective (algebraMap k (C k)) := by
	intro x y h
	have h' := congrArg (factorPoint k) h
	simpa only [AlgHom.commutes, Algebra.algebraMap_self, RingHom.id_apply] using h'

	instance instCharZero [CharZero k] : CharZero (C k) :=
	charZero_of_injective_algebraMap (algebraMap_injective k)

	instance instNontrivial [Nontrivial k] : Nontrivial (C k) :=
	Function.Injective.nontrivial (algebraMap_injective k)

	/-- The first base variable is nonzero in the actual quotient. -/
	theorem a_ne_zero [Nontrivial k] : a k ≠ 0 := by
	intro h
	have := congrArg (factorPoint k) h
	simp at this

	/-- The added base variable is nonzero in the actual quotient. -/
	theorem u_ne_zero [Nontrivial k] : u k ≠ 0 := by
	intro h
	have := congrArg (factorPoint k) h
	simp at this

	/-- The two factors of the highest auxiliary part of `v` are nonzero. -/
	theorem au_sub_d_ne_zero [Nontrivial k] : a k * u k - d k ≠ 0 := by
	intro h
	have := congrArg (factorPoint k) h
	simp at this

	theorem au_add_d_ne_zero [Nontrivial k] : a k * u k + d k ≠ 0 := by
	intro h
	have := congrArg (factorPoint k) h
	simp at this

	/-- Every element of the determinant ring has a polynomial representative. -/
	theorem mk_surjective : Function.Surjective (mk k) :=
	Ideal.Quotient.mk_surjective

	/-- The five canonical quotient variables generate the full algebra. -/
	theorem adjoin_range_generator_via_quotient :
	Algebra.adjoin k (Set.range (generator k)) = ⊤ := by
	have h : (Algebra.adjoin k (Set.range (X : Fin 5 → P k))).map (mk k) = ⊤ := by
	rw [MvPolynomial.adjoin_range_X, Algebra.map_top]
	exact (AlgHom.range_eq_top (mk k)).mpr (mk_surjective k)
	change Algebra.adjoin k (Set.range fun i : Fin 5 => mk k (X i)) = ⊤
	simpa only [AlgHom.map_adjoin, ← Set.range_comp, Function.comp_def] using h

	/-- A derivation is determined by its five generator values. -/
	theorem derivation_eq_zero (D : Derivation k (C k) (C k))
	(ha : D (a k) = 0) (hd : D (d k) = 0) (hb : D (b k) = 0)
	(hc : D (c k) = 0) (hu : D (u k) = 0) : D = 0 := by
	apply DFunLike.ext
	intro z
	obtain ⟨p, rfl⟩ := mk_surjective k z
	change D (mk k p) = 0
	induction p using MvPolynomial.induction_on with
	\| C r =>
	change D (algebraMap k (C k) r) = 0
	exact D.map_algebraMap r
	\| add p q hp hq => simp only [map_add, hp, hq, add_zero]
	\| mul_X p i hp =>
	rw [map_mul, D.leibniz, hp, smul_zero, add_zero]
	have hi : D (mk k (X i)) = 0 := by
	fin_cases i
	· exact ha
	· exact hd
	· exact hb
	· exact hc
	· exact hu
	rw [hi, smul_zero]

	/-- The coefficient of a nonzero derivation fixing `a,d,u` is nonzero. -/
	theorem coefficient_ne_zero (D : Derivation k (C k) (C k))
	(hD : D ≠ 0) (ha : D (a k) = 0) (hd : D (d k) = 0) (hu : D (u k) = 0) :
	rigidityCoefficient D (b k) (c k) ≠ 0 := by
	intro hq
	obtain ⟨hb, hc⟩ := derivation_rigidityCoefficient D (a k) (d k) (b k) (c k)
	(determinant_relation k) ha hd
	rw [hq, mul_zero] at hb hc
	exact hD (derivation_eq_zero k D ha hd hb hc hu)

	/-- The difference-of-squares argument instantiated on the actual quotient. -/
	theorem fixed_of_difference_of_squares [Nontrivial k] [IsDomain (C k)] [CharZero (C k)]
	(D : Derivation k (C k) (C k)) (hD : IsLocallyNilpotent D)
	(hv : D (a k ^ 2 * u k ^ 2 - d k ^ 2) = 0) :
	D (a k) = 0 ∧ D (d k) = 0 ∧ D (u k) = 0 :=
	lnd_fixed_of_difference_of_squares D hD (a_ne_zero k) (u_ne_zero k)
	(au_sub_d_ne_zero k) (au_add_d_ne_zero k) hv

	end StableCoordinate.DeterminantRing

	end

	end OAI